Problem: Let $y=\csc(x)$. What is the value of $\dfrac{dy}{dx}$ at $x=\dfrac{3\pi}{4}$ ? Choose 1 answer: Choose 1 answer: (Choice A) A $\dfrac{1}{\sqrt{2}}$ (Choice B) B $1$ (Choice C) C $-1$ (Choice D) D $\sqrt{2}$
Solution: Let's first find $\dfrac{dy}{dx}$. Then, we can evaluate it at $x=\dfrac{3\pi}{4}$. Recall that the derivative of $\csc(x)$ is $-\dfrac{\cos(x)}{\sin^2(x)}$, or $-\csc(x)\cot(x)$. Put another way, $\dfrac{d}{dx}[\csc(x)]=-\dfrac{\cos(x)}{\sin^2(x)}=-\csc(x)\cot(x)$. [Is there a way to know this without memorizing?] Now let's plug in $x={\dfrac{3\pi}{4}}$ : $\begin{aligned} &\phantom{=}-\dfrac{\cos\left({\dfrac{3\pi}{4}}\right)}{\sin^2\left({\dfrac{3\pi}{4}}\right)} \\\\ &=-\dfrac{-\dfrac{\sqrt2}{2}}{\left(\dfrac{\sqrt{2}}{2}\right)^2} \\\\ &={\dfrac{\sqrt2}{2}}\cdot{\left(\dfrac{4}{2}\right)} \\\\ &=\sqrt2 \end{aligned}$ In conclusion, the value of $\dfrac{dy}{dx}$ at $x=\dfrac{3\pi}{4}$ is $\sqrt2$.